Therefore, I was wondering that how can I apply weight to use with the variance based on logarithmic returns in order to compute the portfolio variance? This is because the modern portfolio theory using simple weight multiply with variance is based on linear returns.

$\begingroup$Logarithmic returns are used in some fields, for example in Option Theory, but in Modern Portfolio Theory simple returns (aka arithmetic returns) are always used. I suggest you switch to using simple returns when working with portfolios.$\endgroup$
– Alex CApr 3 at 1:17